An adjunction formula for the Emerton–Jacquet functor



The Emerton–Jacquet functor is a tool for studying locally analytic representations of p-adic Lie groups. It provides a way to access the theory of p-adic automorphic forms. Here we give an adjunction formula for the Emerton–Jacquet functor, relating it directly to locally analytic inductions, under a strict hypothesis that we call non-critical. We also further study the relationship to socles of principal series in the non-critical setting.


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  1. [BC14]
    J. Bergdall and P. Chojecki, Ordinary representations and companion points for U(3) in the indecomposable case, Preprint (2014).Google Scholar
  2. [BE10]
    C. Breuil and M. Emerton, Représentations p-adiques ordinaires de GL 2(Q p) et compatibilité local-global, Astérisque (2010), 255–315.Google Scholar
  3. [BH15]
    C. Breuil and F. Herzig, Ordinary representations of G(Q p) and fundamental algebraic representations, Duke Math. J., 164 (2015), 1271–1352.MathSciNetCrossRefMATHGoogle Scholar
  4. [BHS17]
    C. Breuil, E. Hellmann and B. Schraen, Une interprétation modulaire de la variété trianguline, Math. Ann., 367 (2017), 1587–1645.MathSciNetCrossRefMATHGoogle Scholar
  5. [Boy99]
    P. Boyer, Mauvaise réduction des variétés de Drinfeld et correspondance de Langlands locale, Invent. Math., 138 (1999), 573–629.MathSciNetCrossRefMATHGoogle Scholar
  6. [Bre15]
    C. Breuil, Vers le socle localement analytique pour GL n II, Math. Ann. 361 (2015), 741–785.MathSciNetCrossRefMATHGoogle Scholar
  7. [Bre16a]
    C. Breuil, Ext1 localement analytique et compatibilité local-global, Preprint (2016).Google Scholar
  8. [Bre16b]
    C. Breuil, Socle localement analytique I, Ann. Inst. Fourier (Grenoble) 66 (2016), 633–685.MathSciNetCrossRefMATHGoogle Scholar
  9. [BZ77]
    I. N. Bernstein and A. V. Zelevinsky, Induced representations of reductive p-adic groups. I, Ann. Sci. École Norm. Sup. (4) 10 (1977), 441–472.MathSciNetCrossRefMATHGoogle Scholar
  10. [CE12]
    F. Calegari and M. Emerton, Completed cohomology—a survey, in Non-abelian fundamental groups and Iwasawa theory, London Math. Soc. Lecture Note Ser., Vol. 393, Cambridge Univ. Press, Cambridge, 2012, pp. 239–257.MATHGoogle Scholar
  11. [CM98]
    R. Coleman and B. Mazur, The eigencurve, in Galois representations in arithmetic algebraic geometry (Durham, 1996), London Math. Soc. Lecture Note Ser., Vol. 254, Cambridge Univ. Press, Cambridge, 1998, pp. 1–113.Google Scholar
  12. [Din14]
    Y. Ding, Formes modulaires p-adiques sur les courbes de Shimura unitaires et comparibilité local-global, Ph.D. thesis, Université Paris-Sud (2014).Google Scholar
  13. [Eme]
    M. Emerton, Jacquet modules of locally analytic representations of p-adic reductive groups. II. The relation to parabolic induction, J. Institut Math. Jussieu., to appear.Google Scholar
  14. [Eme06a]
    M. Emerton, Jacquet modules of locally analytic representations of p-adic reductive groups. I. Construction and first properties, Ann. Sci. École Norm. Sup. (4), 39 (2006), 775–839.MathSciNetCrossRefMATHGoogle Scholar
  15. [Eme06b]
    M. Emerton, On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms, Invent. Math., 164 (2006), 1–84.MathSciNetCrossRefMATHGoogle Scholar
  16. [Eme17]
    M. Emerton, Locally analytic vectors in representations of locally p-adic analytic groups, Mem. Amer. Math. Soc. 248 (2017), iv+158.Google Scholar
  17. [HL11]
    R. Hill and D. Loeffler, Emerton’s Jacquet functors for non-Borel parabolic subgroups, Doc. Math., 16 (2011), 1–31.MathSciNetMATHGoogle Scholar
  18. [Hum08]
    J. E. Humphreys, Representations of semisimple Lie algebras in the BGG category O, Graduate Studies in Mathematics, Vol. 94, American Mathematical Society, Providence, RI, 2008.Google Scholar
  19. [Jac75]
    H. Jacquet, Sur les représentations des groupes réductifs p-adiques, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), Aii, A1271–A1272.Google Scholar
  20. [Jan03]
    J. C. Jantzen, Representations of algebraic groups, second ed., Mathematical Surveys and Monographs, Vol. 107, American Mathematical Society, Providence, RI, 2003.Google Scholar
  21. [OS14]
    S. Orlik and M. Strauch, Category O and locally analytic representations, Preprint (2014).Google Scholar
  22. [OS15]
    S. Orlik and M. Strauch, On Jordan-Hölder series of some locally analytic representations, J. Amer. Math. Soc., 28 (2015), 99–157.MathSciNetCrossRefMATHGoogle Scholar
  23. [Sch11]
    B. Schraen, Représentations localement analytiques de GL3(Qp), Ann. Sci. Éc. Norm. Supér. (4), 44 (2011), 43–145.MathSciNetCrossRefMATHGoogle Scholar
  24. [ST02]
    P. Schneider and J. Teitelbaum, Locally analytic distributions and p-adic representation theory, with applications to GL 2, J. Amer. Math. Soc., 15 (2002), 443–468.MathSciNetCrossRefMATHGoogle Scholar
  25. [ST05]
    P. Schneider and J. Teitelbaum, Duality for admissible locally analytic representations, Represent. Theory, 9 (2005), 297–326.MathSciNetCrossRefMATHGoogle Scholar
  26. [STP01]
    P. Schneider, J. Teitelbaum and D. Prasad, U(g)-finite locally analytic representations, Represent. Theory 5 (2001), 111–128, With an appendix by Dipendra Prasad.MathSciNetCrossRefMATHGoogle Scholar

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© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsBoston UniversityBostonUSA
  2. 2.Mathematical InstituteUniversity of Oxford, Andrew Wiles Building, Radcliffe Observatory QuarterOxfordEngland

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